Semi-Transitive Split Graphs

• Semi-transitive split graphs are defined as split graphs that admit acyclic orientations avoiding forbidden shortcuts, ensuring they are word-representable.
• They are characterized through detailed interval labelling and matrix-theoretic (I-circular) approaches that link combinatorial structures to forbidden subgraph configurations.
• Efficient recognition algorithms using PQ-/PC-tree techniques enable polynomial-time verification of semi-transitivity in split graphs.

A semi-transitive split graph is a split graph that admits a semi-transitive orientation—an acyclic orientation that avoids forbidden shortcut structures as formalized below. This class coincides with the split graphs that are word-representable. The recognition, structural characterization, and forbidden subgraph descriptions of this family have been systematically developed through the interplay of syntactic (word), combinatorial (forbidden subgraph), and matrix-theoretic approaches. The topic is central to combinatorics on words and structural graph theory, specifically in the study of word-representable graph classes.

1. Fundamental Notions

A split graph is a finite simple graph $G=(V,E)$ whose vertex set $V$ can be partitioned as $V = C \cup I$, where $C$ induces a clique and $I$ induces an independent set. By convention, $C$ is taken maximal: no vertex in $I$ is adjacent to all of $C$.

A semi-transitive orientation of an undirected graph $G$ is an orientation of its edges such that:

• the resulting digraph is acyclic,
• for every directed path $u_1 \to u_2 \to \cdots \to u_t$ ($t \ge 2$), it holds that either $(u_1, u_t) \notin E$ (no shortcut) or all intermediate arcs $u_i \to u_j$ for $1 \le i < j \le t$ are present (total shortcut).

A split graph $G$ is semi-transitive if it admits a semi-transitive orientation. This property is equivalent to word-representability, i.e., $G$ can be encoded by a word over its vertex set such that adjacency corresponds to alternation in the word (Srinivasan et al., 13 Dec 2025, Kitaev et al., 2017).

2. Structural Characterizations

2.1 Vertex Neighborhoods and Interval Structure

A core structural result states that $G=(C \cup I, E)$ is semi-transitive if and only if there is a linear labelling $C = \{1,2,\dots,k\}$ such that for every $a \in I$, its neighborhood $N(a) \subseteq C$ is either:

• a single interval $[b,c]$, or
• the union of two end-intervals $[1,b] \cup [c,k]$ with $b < c$,

and every pair of such interval types from $I$ satisfies certain strict non-overlapping/crossing conditions. These interval and crossing conditions prevent the existence of forbidden shortcuts (Kitaev et al., 2021, Dwary et al., 2 Feb 2025, Kitaev et al., 2017).

2.2 Matrix-Theoretic (I-Circular Property)

Given the $|I| \times |C|$ bipartite adjacency matrix $A(G)$ of the split graph (rows indexed by $I$, columns by $C$), $G$ is semi-transitive if and only if $A(G)$ has the I-circular property: there is a column ordering such that:

• each row's 1's appear in a circular interval,
• and for every pair of rows, the common 1's also form a circular interval.

This property extends classical consecutive-ones notions, linking semi-transitive orientability to forbidden submatrix configurations and enabling polynomial time recognition (Srinivasan et al., 13 Dec 2025).

3. Recognition Complexity and Algorithms

• General graphs: Deciding semi-transitive orientability is NP-complete (Kitaev et al., 2021).
• Split graphs: Recognition is polynomial-time solvable. Specifically, it can be decided in $\mathcal{O}(|I|^2 |C|)$ time. The algorithm constructs suitable biadjacency/circular-ones matrices and applies PQ-/PC-tree techniques to test for required interval properties (Kitaev et al., 2021, Srinivasan et al., 13 Dec 2025).
• Infinite families and morphisms: For directed split graphs generated by matrix morphisms, semi-transitivity is classified entirely in terms of the row patterns and forbidden "interlacing" rows in the generating matrices (Iamthong et al., 2021).

4. Forbidden Induced Subgraphs and Submatrices

A complete forbidden submatrix characterization of the I-circular property translates to a forbidden induced subgraph characterization for semi-transitive split graphs (Srinivasan et al., 13 Dec 2025). The minimal forbidden configurations fall into infinite and sporadic families:

Matrix Family / Graph Family	Description	Parameterization
Cycle matrices $M_k, \bar M_k$	$k$-cycle of neighborhoods	$k\ge3$
$M_k^*, \bar M_k^*$	Cycle plus isolated vertex	$k\ge3$
Four small bracelets / $\bar A_4^j$	Small, specific $3\times 4$ submatrices	$k=4$
$B$, $\bar B$	Small configurations of size $3\times 4$	fixed
Infinite split graphs S₁, S₂, S₃	Clique $K_k$ plus independent set attached to consecutive blocks, zig-zag, or complementary pairs	$k\ge3,4$
Six sporadic graphs	As induced by small forbidden matrices	$|C|=4,5$, $|I|=3,4$

Each forbidden submatrix encodes an induced split subgraph where any orientation necessarily yields a forbidden shortcut or cycle.

For small parameter cases, (Kitaev et al., 2021, Kitaev et al., 2017) established explicit lists:

• For $|I|\leq 3$, exactly three minimal forbidden induced subgraphs.
• For $|C|\leq 4,5$, explicit finite lists of forbidden graphs (4 and 9, respectively).
• For the degree $\leq 2$ case on $I$, explicit infinite families $T_2$ and $A_\ell$.

5. Algorithms and Matrix Reformulations

Recognition proceeds by reducing the problem to testing the consecutive-ones or circular-ones property in matrices derived from the bipartite adjacency structure. Given $G=(C \cup I, E)$:

• Construct the matrix where each row is an $I$-vertex, and each column a $C$-vertex. Populate 1's for adjacency.
• Use circular-ones property algorithms (PQ-tree, PC-tree) to test for a permissible row/column ordering.
• Absence of any forbidden submatrix from the family $𝒥$ implies $G$ is semi-transitive (Srinivasan et al., 13 Dec 2025).

For orientation, one induces a tournament on $C$ consistent with the column order, orients $I \to C$ (or $C \to I$) edges following the intervals described above.

6. Extension to Infinite and Morphic Split Graphs

Infinite split graphs generated by iterative matrix morphisms have been completely classified as semi-transitive via the row-type patterns and overlap constraints in their seed matrices. Only matrices whose rows are of the forms $0^r 1^s 0^t$, $0^r (-1)^s 0^t$, or $1^r 0^s (-1)^t$, and that avoid forbidden row interlacing, ever yield infinite semi-transitive split graphs under this construction (Iamthong et al., 2021).

7. Representation Number and Quantitative Extremes

Every semi-transitive split (word-representable split) graph has representation number at most $3$, i.e., can be encoded by a $3$-uniform word. The extremal graphs requiring three are precisely those containing certain even $k$-sun graphs or the infinite $F_0, F_1(k), F_2(k)$ families as induced subgraphs. The boundary between representation number $2$ (circle graphs) and $3$ has been sharply characterized (Dwary et al., 2 Feb 2025).

• S. Kitaev, A. Pyatkin, "On semi-transitive orientability of split graphs" (Kitaev et al., 2021)
• B. Srinivasan, A. Hariharasubramanian, "Forbidden Induced Subgraph Characterization of Word-Representable Split Graphs" (Srinivasan et al., 13 Dec 2025)
• S. Kitaev, Y. Long, Z. Ma, Y. Wu, "Word-representability of split graphs" (Kitaev et al., 2017)
• S. Kitaev, A. Pyatkin, "Semi-transitivity of directed split graphs generated by morphisms" (Iamthong et al., 2021)
• T. Dwary, T. Mozhui, C. Krishna, "Representation Number of Word-Representable Split Graphs" (Dwary et al., 2 Feb 2025)